Set In Rn Research Articles

Weak Harnack inequality for doubly non-linear equations of slow diffusion type

We consider non-negative weak super-solutions u:ΩT→R≥0 to the doubly non-linear equation∂t(|u|q−1u)−divA(x,t,u,Du)=0inΩT=Ω×(0,T], where Ω is an bounded open set in RN for N≥2, T>0 and q is a non-negative parameter. Furthermore, the vector field A satisfies standard p-growth assumptions for some p>1. The main novelty of this paper is that we establish the weak Harnack inequality in the entire slow diffusion regime p−q−1>0. Additionally, we only require that the weak super-solution u is located in the function spaceCloc0([0,T];Llocq+1(Ω))∩Llocp(0,T;Wloc1,p(Ω)).

A Variational Surface-Evolution Approach to Optimal Transport over Transitioning Compact Supports with Domain Constraints

We examine the optimal mass transport problem in Rn between densities with transitioning compact support by considering the geometry of a continuous interpolating support boundary Γ in space-time within which the mass density evolves according to the fluid dynamical framework of Benamou and Brenier. We treat the geometry of this space-time embedding in terms of points, vectors, and sets in Rn+1=R×Rn and blend the mass density and velocity as well into a space-time solenoidal vector field W|Ω→Rn+1 over a compact set Ω⊂Rn+1. We then formulate a joint optimization for W and its support using the shaped gradient of the space-time surface Γ outlining the support boundary ∂Ω. This easily accommodates spatiotemporal constraints, including obstacles or mandatory regions to visit.

Generalized fuzzy difference method for solving fuzzy initial value problem

We are going to explain the fuzzy Adams–Bashforth methods for solving fuzzy differential equations focusing on the concept of g-differentiability. Considering the analysis of normal, convex, upper semicontinuous, compactly supported fuzzy sets in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R^n$$\\end{document} and also convergence of the methods, the general expression of solutions is obtained. Finally, we demonstrate the importance of our method with some illustrative examples. These examples are provided aiming to solve the fuzzy differential equations.

Existence and symmetry of periodic nonlocal-CMC surfaces via variational methods

Abstract This paper provides the first variational proof of the existence of periodic nonlocal-CMC surfaces. These are nonlocal analogues of the classical Delaunay cylinders. More precisely, we show the existence of a set in R n \mathbb{R}^{n} which is periodic in one direction, has a prescribed (but arbitrary) volume within a slab orthogonal to that direction, has constant nonlocal mean curvature, and minimizes an appropriate periodic version of the fractional perimeter functional under the volume constraint. We show, in addition, that the set is cylindrically symmetric and, more significantly, that it is even as well as nonincreasing on half its period. This monotonicity property solves an open problem and an obstruction which arose in an earlier attempt, by other authors, to show the existence of minimizers.

Quasilinear problems with nonlinear boundary conditions in higher-dimensional thin domains with corrugated boundaries

AbstractIn this work, we analyze the asymptotic behavior of a class of quasilinear elliptic equations defined in oscillating(N+1)\left(N+1)-dimensional thin domains (i.e., a family of bounded open sets fromRN+1{{\mathbb{R}}}^{N+1}, with corrugated bounder, which degenerates to an open bounded set inRN{{\mathbb{R}}}^{N}). We also allow monotone nonlinear boundary conditions on the rough border whose magnitude depends on the squeezing of the domain. According to the intensity of the roughness and a reaction coefficient term on the nonlinear boundary condition, we obtain different regimes establishing effective homogenized limits inNN-dimensional open bounded sets. In order to do that, we combine monotone operator analysis techniques and the unfolding method used to deal with asymptotic analysis and homogenization problems.

Parameter estimates and a uniqueness result for double phase problem with a singular nonlinearity

We consider the boundary value problem −Δpuλ−Δquλ=λg(x)uλ−β in Ω, uλ=0 on ∂Ω with uλ>0 in Ω. We assume Ω is a bounded open set in RN with smooth boundary, 1<p<q<∞, β∈[0,1), g is a positive weight function and λ is a positive parameter. We derive an estimate for uλ which describes its exact behavior when the parameter λ is large. In general, by invoking appropriate comparison principles, this estimate can be used as a powerful tool in deducing the existence, non-existence and multiplicity of positive solutions of nonlinear elliptic boundary value problems. Here, as an application of this estimate, we obtain a uniqueness result for a nonlinear elliptic boundary value problem with a singular nonlinearity.

Minimax decision rules for planning under uncertainty: Drawbacks and remedies

It is common to use minimax rules to make planning decisions when there is great uncertainty about what may happen in the future. Using minimax rules avoids the need to determine probabilities for each future scenario, which is an attractive feature in many public sector settings. However there are potential problems in the application of a minimax approach. In this paper our aim is to give guidance for planners considering a minimax approach, including minimax regret which is one popular version of this. We give an analysis of the behaviour of minimax rules in the case with a finite set of possible future scenarios. Minimax rules will have sensitivity to the choice of a small number of scenarios. When regret-based rules are used there are also problems arising since the independence of irrelevant alternatives property fails, which can lead to opportunities to game the process. We analyse these phenomena considering cases where the decision variables are chosen from a convex set in Rn, as well as cases with a finite set of decision choices. We show that the drawbacks of minimax regret hold even when restrictions are placed on the problem setup, and we show how working with a structured set of scenarios can ameliorate the difficulty of having a final decision depend on the characteristics of just a handful of extreme scenarios.

Explicit Salem sets in [formula omitted

We construct the first explicit (i.e., non-random) examples of Salem sets in Rn of arbitrary prescribed Hausdorff dimension. This completely resolves a problem proposed by Kahane more than 50 years ago. The construction is based on a form of Diophantine approximation in number fields.

Analyticity and observability for fractional order parabolic equations in the whole space

In this paper, we study the quantitative analyticity and observability inequality for solutions of fractional order parabolic equations with space-time dependent potentials in ℝn. We first obtain a uniformly lower bound of analyticity radius of the spatial variable for the above solutions with respect to the time variable. Next, we prove a globally Hölder-type interpolation inequality on a thick set, which is based on a propagation estimate of smallness for analytic functions. Finally, we establish an observability inequality from a thick set in ℝn, by utilizing a telescoping series method.

Convergence of weak solutions of elliptic problems with datum in L 1

Motivated by the $Q$-condition result proven by Arcoya and Boccardo in [J. Funct. Anal. 268(2015), No. 5, 1153–1166], we analyze the behaviour of the weak solutions { − Δ p u ε + ε | f ( x ) | u ε = f ( x ) in Ω , u ε = 0 on ∂ Ω , when ε tends to 0 . Here, Ω denotes a bounded open set of R N ( N ≥ 2 ) , − Δ p u = − d i v ( | ∇ u | p − 2 ∇ u ) is the usual p -Laplacian operator ( 1 &lt; p &lt; ∞ ) and f ( x ) is an L 1 ( Ω ) function. We show that this sequence converges in some sense to u , the entropy solution of the problem { − Δ p u = f ( x ) in Ω , u = 0 on ∂ Ω . In the semilinear case, we prove stronger results provided the weak solution of that problem exists.

Lipschitz regularity of minimizers of variational integrals with variable exponents

In this paper we prove the Lipschitz regularity for local minimizers of convex variational integrals of the form F(v,Ω)=∫ΩF(x,Dv(x))dx,where, for n≥2 and N≥1, Ω is a bounded open set in Rn, u∈W1,1(Ω,RN) and the energy density F:Ω×RN×n→R satisfies the so called variable growth conditions. The main novelty here is that we assume an almost critical regularity in the Orlicz Sobolev setting for the energy density as a function of the x variable.

Asymptotic dimension of intersection graphs

We show that intersection graphs of compact convex sets in Rn of bounded aspect ratio have asymptotic dimension at most 2n+1. More generally, we show this is the case for intersection graphs of systems of subsets of any metric space of Assouad–Nagata dimension n that satisfy the following condition: For each r,s>0 and every point p, the number of pairwise-disjoint elements of diameter at least s in the system that are at distance at most r from p is bounded by a function of r/s.

On closed weakly m-convexsets

In the present work we study properties of generally convex sets in the n-dimensional real Euclidean space Rn, (n&gt;1), known as weakly m-convex, m=1,...,n-1. An open set of Rn is called weakly m-convex if, for any boundary point of the set, there exists an m-dimensional plane passing through this point and not intersecting the given set. A closed set of Rn is called weakly m-convex if it is approximated from the outside by a family of open weakly m-convex sets. A point of the complement of a set of Rn to the whole space is called an m-nonconvexity point of the set if any m-dimensional plane passing through the point intersects the set. It is proved that any closed, weakly (n-1)-convex set in Rn with non-empty set of (n-1)-nonconvexity points consists of not less than three connected components. It is also proved that the interior of a closed, weakly 1-convex set with a finite number of components in the plane is weakly 1-convex. Weakly m-convex domains and closed connected sets in Rn with non-empty set of m-nonconvexity points are constructed for any n&gt;2 and any m=1,...,n-1.

The topological structure of the set of fuzzy numbers with Lp metric

For a non-degenerate convex subset Y of the n-dimensional Euclidean space Rn, let K(Y) be the family of all fuzzy sets in Rn, which are upper-semicontinuous, fuzzy convex, normal and with compact supports included in Y. We show that the space K(Y) with the topology of the Lp metric with p≥1, that is, for f,g∈K(Y),Lp(f,g)=(∫01(dH([f]α,[g]α))pdα)1p, is homeomorphic to the Hilbert cube Q=[−1,1]ω if and only if Y is compact; and the space K(Y) is homeomorphic to the pseudoboundary B(Q)=Q∖(−1,1)ω of the Hilbert cube Q=[−1,1]ω if and only if Y is non-compact and locally compact.

Poincaré inequalities and compact embeddings from Sobolev type spaces into weighted Lq spaces on metric spaces

We study compactness and boundedness of embeddings from Sobolev type spaces on metric spaces into Lq spaces with respect to another measure. The considered Sobolev spaces can be of fractional order and some statements allow also nondoubling measures. Our results are formulated in a general form, using sequences of covering families and local Poincaré type inequalities. We show how to construct such suitable coverings and Poincaré inequalities. For locally doubling measures, we prove a self-improvement property for two-weighted Poincaré inequalities, which applies also to lower-dimensional measures.We simultaneously treat various Sobolev spaces, such as the Newtonian, fractional Hajłasz and Poincaré type spaces, for rather general measures and sets, including fractals and domains with fractal boundaries. By considering lower-dimensional measures on the boundaries of such domains, we obtain trace embeddings for the above spaces. In the case of Newtonian spaces we exactly characterize when embeddings into Lq spaces with respect to another measure are compact. Our tools are illustrated by concrete examples. For measures satisfying suitable dimension conditions, we recover several classical embedding theorems on domains and fractal sets in Rn.

Principal kinematic formulas for germs of closed definable sets

We prove two principal kinematic formulas for germs of closed definable sets in Rn, that generalize the Cauchy-Crofton formula for the density due to Comte and the infinitesimal linear kinematic formula due to the author. In this setting, we do not integrate on the space of euclidean motions SO(n)⋉Rn, but on the manifold SO(n)×Sn−1.

Topological properties of closed weakly m-semiconvex sets

The present work considers properties of generally convex sets in the n-dimensional real Euclidean space ℝn, n > 1, known as weakly m-semiconvex, m = 1, 2, … , n − 1. For all that, the subclass of not m-semiconvex sets is distinguished from the class of weakly m-semiconvex sets. A set of the space ℝn is called m-semiconvex if, for any point of the complement of the set to the whole space, there is an m-dimensional half-plane passing through this point and not intersecting the set. An open set of ℝn is called weakly m-semiconvex if, for any point of the boundary of the set, there exists an m-dimensional half-plane passing through this point and not intersecting the given set. A closed set of ℝn is called weakly m-semiconvex if it is approximated from the outside by a family of open weakly m-semiconvex sets. An example of a closed set with three connected components of the subclass of weakly 1-semiconvex but not 1-semiconvex sets in the plane is constructed. It is proved that this number of components is minimal for any closed set of the subclass. An example of a closed set of the subclass with a smooth boundary and four components is constructed. It is proved that this number of components is minimal for any closed, bounded set of the subclass having a smooth boundary and a not 1-semiconvex interior. It is also proved that the interior of a closed, weakly 1-semiconvex set with a finite number of components in the plane is weakly 1-semiconvex. Weakly m-semiconvex but not m-semiconvex domains and closed connected sets in ℝn are constructed for any n ≥ 3 and any m = 1, 2, … , n − 2.

High-dimensional asymptotics of likelihood ratio tests in the Gaussian sequence model under convex constraints

In the Gaussian sequence model Y=μ+ξ, we study the likelihood ratio test (LRT) for testing H0:μ=μ0 versus H1:μ∈K, where μ0∈K, and K is a closed convex set in Rn. In particular, we show that under the null hypothesis, normal approximation holds for the log-likelihood ratio statistic for a general pair (μ0,K), in the high-dimensional regime where the estimation error of the associated least squares estimator diverges in an appropriate sense. The normal approximation further leads to a precise characterization of the power behavior of the LRT in the high-dimensional regime. These characterizations show that the power behavior of the LRT is in general nonuniform with respect to the Euclidean metric, and illustrate the conservative nature of existing minimax optimality and suboptimality results for the LRT. A variety of examples, including testing in the orthant/circular cone, isotonic regression, Lasso and testing parametric assumptions versus shape-constrained alternatives, are worked out to demonstrate the versatility of the developed theory.

An improvement of Bernstein’s inequality for functions in Orlicz spaces with smooth Fourier image

Let Φ:[0,+∞)→[0,+∞] be an arbitrary Young function and K be a compact set in ℝn having (O)-property. Then there exists a constant CK,Φ<∞ independent of f such that ∥Dαf∥(Φ)≤CK,Φsupz∈K|zα|∥f∥ℒK,3 for all α∈ℤ+n and f∈ℒK,3, where ℒK,3={f∈𝒮′(ℝn):suppf^⊂K,D(3,3,…,3)f^∈C(ℝn)}, ∥f∥ℒK,3=∥D(3,3,…,3)f^∥∞, f^ is the Fourier transform of f and ∥⋅∥(Φ) is the Luxemburg norm. As an application, a new Paley–Wiener theorem is given.

The restricted content and the d-dimensional Analyst's Travelling Salesman Theorem for general sets

In his 1990 paper, Jones characterized subsets of rectifiable curves in R2 via a multiscale sum of β-numbers, which measure how far a given set deviates from a straight line at each scale and location. This characterization was extended by Okikiolu to subsets of Rn and by Schul to subsets of a Hilbert space.Recently, there has been some interest in characterizing subsets of higher dimensional surfaces in Rn. Using a variant of Jones' β-number introduced by Azzam and Schul, Villa gave a characterization of lower regular subsets of a certain class of topologically stable surfaces – introduced in a 2004 paper of David – via a multiscale sum of these new β-numbers.In this paper we remove the lower regularity condition and prove an analogous result for general d-dimensional subsets of Rn. To do this, we introduce the restricted content, which assigns ‘mass’ to any subset of Rn (even to sets with zero Hausdorff measure), and use it to define new d-dimensional variant of Jones' β-number that is defined for any set in Rn.